Method and apparatus for minimizing unwanted dynamics in a physical system
First Claim
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1. A method for controlling a physical system by generating an input to the system to minimize unwanted dynamics in the system response comprising;
- establishing expressions quantifying the unwanted dynamics;
establishing first constraints bounding an available input to the dynamic system;
establishing second constraints bounding the unwanted dynamics;
finding a solution which allows maximum variations in physical system characteristics and is used to generate the input while still satisfying the first and second constraints; and
controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized.
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Abstract
Method and apparatus for minimizing unwanted dynamics in a physical system response. Constraints on the input and on the unwanted dynamics are established, and an impulse sequence which satisfies the constraints is determined. The impulse sequence is convolved with an arbitrary command input to produce a shaped input which is used to drive physical system, thereby minimizing unwanted dynamics. The constraints on the input and on the unwanted dynamics may be selected to achieve shaped inputs and residual unwanted dynamics of various characteristics.
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Citations
130 Claims
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1. A method for controlling a physical system by generating an input to the system to minimize unwanted dynamics in the system response comprising;
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establishing expressions quantifying the unwanted dynamics; establishing first constraints bounding an available input to the dynamic system; establishing second constraints bounding the unwanted dynamics; finding a solution which allows maximum variations in physical system characteristics and is used to generate the input while still satisfying the first and second constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18)
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14. The method of claim 13 wherein the sequence of impulses is based on an approximate solution to the first set of equations.
- 15. The method of claim 14 wherein the number of impulses is three and the approximate solution is
- space="preserve" listing-type="equation">A.sub.1 =0.2497+0.2496V+0.8001ζ
+1.233Vζ
+0.4960ζ
.sup.2 +3.173Vζ
.sup.2
space="preserve" listing-type="equation">A.sub.2 =1-(A.sub.1 +A.sub.3)
space="preserve" listing-type="equation">A.sub.3 =0.2515+0.2147V-0.8325ζ
+1.415Vζ
+0.8518ζ
.sup.2 -4.901Vζ
.sup.2
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.5000+0.4616Vζ
+4.262Vζ
.sup.2 +1.756Vζ
.sup.3 +8.578V.sup.2 ζ
-108.6V.sup.2 ζ
.sup.2 +337.0V.sup.2 ζ
.sup.3)T.sub.dt3 =Td where ##EQU54## - space="preserve" listing-type="equation">A.sub.1 =0.2497+0.2496V+0.8001ζ
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- 16. The method of claim 14 wherein the number of impulses is four, V=0.05 and the approximate solution is
- space="preserve" listing-type="equation">A.sub.1 =0.1608+0.7475ζ
+1.948ζ
.sup.2 -0.4882ζ
.sup.3
space="preserve" listing-type="equation">A.sub.2 =1-(A.sub.1 +A.sub.3 +A.sub.4)
space="preserve" listing-type="equation">A.sub.3 =0.3394-0.5466ζ
-1.1354ζ
.sup.2 +2.6167ζ
.sup.3
space="preserve" listing-type="equation">A.sub.4 =0.1589-0.5255ζ
+0.4152ζ
.sup.2 +1.0164ζ
.sup.3
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.5000+0.1426ζ
-0.6243ζ
.sup.2 +6.590ζ
.sup.3)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(1.0+0.17226ζ
-1.725ζ
.sup.2 +10.058ζ
.sup.3)T.sub.d
space="preserve" listing-type="equation">t.sub.4 =T.sub.dwhere ##EQU55## - space="preserve" listing-type="equation">A.sub.1 =0.1608+0.7475ζ
- and a damping ratio ζ
=0;the solution is a sequence of four impulses of amplitude Ai and location ti ; the expression quantifying the unwanted dynamics is
space="preserve" listing-type="equation">R(Ω
)={[Σ
A.sub.i cos (Ω
t.sub.i)].sup.2 +[Σ
A.sub.i sin (Ω
t.sub.i)].sup.2 }.sup.1/2where R(Ω
) is the ratio of the residual vibration produced by the sequence of impulses to the residual vibration produced by a single impulse of amplitude Σ
Ai, if the physical system had a natural frequency Ω
;the second constraint comprises selecting a maximum V bounding the residual vibration R(Ω
); andthe four impulses are defined by the equations
space="preserve" listing-type="equation">A.sub.1 =(3×
.sup.2 +2×
+3V.sup.2)/(16×
)
space="preserve" listing-type="equation">A.sub.2 =0.5-(3×
.sup.2 +2×
+3V.sup.2)/(16×
)
space="preserve" listing-type="equation">A.sub.3 =0.5-(3×
.sup.2 +2×
+3V.sup.2)/(16×
)
space="preserve" listing-type="equation">A.sub.4 =(3×
.sup.2 +2×
+3V.sup.2)/(16×
)
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =0.5T.sub.d
space="preserve" listing-type="equation">t.sub.3 =T.sub.d
space="preserve" listing-type="equation">t.sub.4 =1.5T.sub.dwhere ##EQU56## and Td =2π
/ω
-
i and damping ratios ζ
i, i=1 . . . p, establishing p sets of constraint equations with the equations representing the first and second series of constraints, and finding a solution which allows maximum variations in physical system characteristics while still satisfying the p sets of constraints.
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19. A method for controlling a physical system by generating an input to the system to minimize unwanted dynamics in the system response comprising;
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establishing expressions quantifying the unwanted dynamics; establishing first constraints bounding an available input to the dynamic system; establishing second constraints bounding the unwanted dynamics; finding a minimum length solution which is used to generate the input while satisfying the first and second constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54)
- 46. The method of claim 19 wherein the solution is a sequence of an even number of impulses, which amplitude are given by
- space="preserve" listing-type="equation">A.sub.i =(-1).sup.i-1.
-
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47. The method of claim 45 wherein the dynamic system is modelled as M masses, m.sub.,m2, . . . mM, connected by springs, and has p vibrational modes each characterized by an undamped natural frequency ω
-
i and damping ratio ζ
i, i=1 . . . p; and
the locations of the impulses are defined by the equation ##EQU63## where Σ
m is the sum of the M masses and xd is a pre-selected distance to be moved, and the equations ##EQU64## for each of the p modes, where ω and
ζ
are the undamped natural frequency and damping constant for the mode, ti is the location of the ith impulse, t1 has been arbitrarily set to 0, V is the bound on the residual vibrations of the physical system, and ω
hi and ω
low are a frequency above and below the natural frequency ω
for which the residual vibration is reduced to zero.
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i and damping ratio ζ
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48. The method of claim 47 wherein M=2, p=1, and n=7.
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49. The method of claim 19 further comprising:
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establishing third constraints bounding variations in physical system characteristics; finding a minimum length solution which is used to generate the input while satisfying the first, second and third constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized.
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50. The method of claim 49 wherein the solution is a minimum length sequence of impulses which satisfies the first, second and third constraints;
- and finding the minimum length solution comprises first determining the number of impulses required, and then determining the amplitudes and locations of each impulse.
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51. The method of claim 50 wherein determining the number of impulses required comprises the use of a decision tree.
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52. The method of claim 51 wherein the dynamic system is characterized by an undamped natural frequency ω
- and damping ratio ζ
;
the second constraint comprises selecting a bound, V=0.05, on the residual vibrations of the physical system;
the third constraint comprises selecting an insensitivity, I, bounding the variation of ω
; and
the number of impulses is determined by using the following decision tree;
space="preserve" listing-type="equation">if I<
0.06363+0.01044ζ
+0.07064ζ
.sup.2 +0.40815ζ
.sup.3,then the number of impulses=2;
space="preserve" listing-type="equation">if I<
0.3991+0.6313ζ
+0.3559ζ
.sup.2 +2.3052ζ
.sup.3
space="preserve" listing-type="equation">and I>
0.06363+0.01044ζ
+0.07064ζ
.sup.2 +0.40815ζ
.sup.3,then the number of impulses=3;
space="preserve" listing-type="equation">if I>
0.3991+0.6313ζ
+0.3559ζ
.sup.2 +2.3052ζ
.sup.3then the number of impulses=4.
- and damping ratio ζ
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53. The method of claim 49 wherein the dynamic system has a single vibrational mode characterized by an undamped natural frequency ω
- and damping ratio ζ
;
the second constraint comprises bounding the residual vibrations in the physical system; and
the third constraint comprises bounding the variation on ω
.
- and damping ratio ζ
-
54. The method of claim 49 wherein the dynamic system has a single vibrational mode characterized by an undamped natural frequency ω
- and damping ratio ζ
;
the second constraint comprises bounding the residual vibrations in the physical system; and
the third constraint comprises bounding the variation on ζ
.
- and damping ratio ζ
-
55. A method for controlling a physical system by generating an input to the system to minimize unwanted dynamics in the system response comprising;
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establishing expressions quantifying the unwanted dynamics; establishing first constraints bounding an available input to the dynamic system; establishing second constraints on variation in system response with variations in the physical system characteristics; finding a minimum length solution which is used to generate the input while satisfying the first and second constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74)
- 69. The method of claim 66 wherein n=3, P=2, and the locations of the three impulses are approximated by the following equations
- space="preserve" listing-type="equation">t.sub. =0
space="preserve" listing-type="equation">t.sub.2 =(0.12929+0.09393ζ
-0.06204ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(0.20975+0.02418ζ
-0.07474ζ
.sup.2)T.sub.dwhere ##EQU67##
-
- 70. The method of claim 66 wherein n=3, P=3, and the locations of the three impulses are approximated by the following equations
- space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.10089+0.05976ζ
-0.05376ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(0.17420+0.01145ζ
-0.07317ζ
.sup.2)T.sub.dwhere ##EQU68##
-
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.15236+0.23230ζ
+0.09745ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(0.27750+0.10237ζ
-0.00612ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.4 =(0.63139+0.33716ζ
-0.07724ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.5 =(0.67903+0.18179ζ
-0.06008ζ
.sup.2)T.sub.dwhere ##EQU70##
-
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.11700+0.15424ζ
+0.03449ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(0.26041+0.11899ζ
-0.05910ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.4 =(0.49378+0.15092ζ
-0.25380ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.5 =(0.56273+0.04255ζ
-0.19898ζ
.sup.2)T.sub.dwhere ##EQU71##
-
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">t.sub.2 =(0.10022+0.11695ζ
+0.00246ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.3 =(0.24352+0.10877ζ
-0.08790ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.4 =(0.44109+0.11059ζ
-0.23127ζ
.sup.2)T.sub.d
space="preserve" listing-type="equation">t.sub.5 =(0.51155+0.02121ζ
-0.20054ζ
.sup.2)T.sub.dwhere ##EQU72##
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75. A method for controlling a physical system by generating an input to a dynamic system with p vibrational modes characterized by undamped natural frequencies ω
- and damping ratios ζ
i, i=1 . . . p, to minimize unwanted dynamics in the physical system response comprising;defining a sequence of n=2p+1 impulses by the equations ##EQU73## for each of the p vibrational modes, ##EQU74##
space="preserve" listing-type="equation">t.sub.1 =0
space="preserve" listing-type="equation">A.sub.i ≧
0where Ai are the amplitudes of the impulses and ti are the locations of the impulses;
approximating the solution to the defining equations; andcontrolling the physical system based on the generated input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (76)
- and damping ratios ζ
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77. A method for controlling a physical system by generating an input to the system to minimize unwanted dynamics in the system response comprising;
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establishing first constraints on a sequence of impulses which minimize the unwanted dynamics; determining a first sequence of impulses which satisfies the first constraints; determining a second sequence of impulses, which is discretized in locations and is used to generate the input, by determining the number of impulses and the locations of the impulses of the second sequence based on the first sequence; determining the amplitudes of the second sequence to satisfy second constraints based on the first constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (78, 79)
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80. A method for controlling a physical system by generating an input to a dynamic physical system to reduce the deviation between the shape of a trajectory traversed by a point in the physical system and a pre-selected shape comprising:
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establishing constraints on the available inputs to the dynamic system to define a group of possible inputs; determining an impulse sequence which eliminates unwanted dynamics in the physical system; convolving the impulse sequence with each input in the group of possible inputs to determine a group of shaped inputs; determining the shaped input which minimizes the deviation between the shape of the actual trajectory and the pre-selected shape; and controlling the physical system based on the shaped input which minimizes the deviation between the shape of the actual trajectory and the pre-selected shape, whereby the deviation is minimized. - View Dependent Claims (81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94)
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95. A method for controlling a physical system by generating an input to a dynamic system to minimize unwanted dynamics in the physical system response comprising:
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establishing expressions quantifying the unwanted dynamics; establishing first constraints on the available inputs to the dynamic system to define a group of possible inputs; expressing each input in the group of possible inputs as a combination of one or more primitive input trains; determining the combination which minimizes the unwanted dynamics; controlling the physical system based on the input in the group of possible inputs which corresponds to the combination which minimizes the unwanted dynamics, whereby unwanted dynamics are minimized. - View Dependent Claims (96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109)
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110. A method for shaping an arbitrary command input to a dynamic physical system to reduce unwanted dynamics in the physical system comprising:
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determining a first parameterization for the arbitrary command input; determining an impulse sequence which eliminates unwanted dynamics in the physical system; determining a second parameterization to express the convolution of the impulse sequence with the arbitrary command input; and controlling the input to the physical system based on the second parameterization, whereby unwanted dynamics in the physical system are minimized. - View Dependent Claims (111, 112, 113, 114, 115)
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116. An apparatus for controlling a physical system by reducing the deviation between the shape of a trajectory traversed by a point in a dynamic physical system and a pre-selected shape comprising:
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computing apparatus for determining an impulse sequence which eliminates unwanted dynamics in the physical system; apparatus for selecting an input from a group of possible inputs which, when convolved with the impulse sequence, minimizes the deviation between the shape of the actual trajectory and the pre-selected shape; and apparatus for controlling the physical system based on the convolution of the impulse sequence with the selected input whereby the deviation between the shape of the actual trajectory and the pre-selected shape is minimized.
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117. An apparatus for controlling a physical system by shaping an arbitrary command input to a physical system in order to minimize unwanted dynamics in a physical system response comprising:
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a sensor for measuring the unwanted dynamics; computing apparatus for determining natural frequencies of the unwanted dynamics based on the measurement of the unwanted dynamics; apparatus for determining a sequence of impulses, a sequence of pulses, or a trajectory which minimizes the unwanted dynamics at the natural frequencies; apparatus for convolving the sequence of impulses with the arbitrary command input to produce a shaped command input; and apparatus for controlling the physical system based on the shaped command input whereby the unwanted dynamics are minimized.
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118. An apparatus for minimizing unwanted dynamics in a physical system response comprising:
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computing apparatus for expressing each input in a group of possible inputs as a combination of impulse sequences and a step function; apparatus for determining which of the combinations of impulse sequences minimizes the unwanted dynamics; apparatus for controlling the physical system based on the input which corresponds to the combination of impulse sequences which minimizes unwanted dynamics.
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119. Apparatus for controlling a physical system by shaping an input to a dynamic physical system to reduce unwanted dynamics in the physical system comprising:
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apparatus for determining a command which eliminates unwanted dynamics in the physical system; computing apparatus for splitting the command into an impulse sequence and an input such that the command is obtained if the input and the impulse sequence are convolved together; apparatus for convolving the sequence of impulses with an arbitrary command input to produce a shaped output; and apparatus for controlling the physical system based on the shaped input whereby unwanted dynamics are minimized.
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120. An apparatus for reducing unwanted dynamics in a physical system response of a closed loop dynamic system comprising:
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a first filter which is based on an impulse sequence which eliminates unwanted dynamics in the physical system response and which is located in a feed-forward branch of the closed loop system; and a second filter which is based on an inverse of the first filter and which is located in the feed-back branch of the closed loop system, whereby unwanted dynamics are minimized. - View Dependent Claims (121, 122, 123, 124)
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125. A method for reducing unwanted dynamics in a physical system response of a closed loop dynamic system comprising:
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determining an impulse sequence which eliminates unwanted dynamics in the physical system response; inserting a first filter based on the impulse sequence in a feed-forward branch of the closed loop system; and inserting a second filter based on an inverse of the first filter in the feed-back branch of the closed loop system whereby unwanted dynamics are minimized.
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126. A method for controlling a physical system by generating an input to a dynamic system to minimize unwanted dynamics in a physical system response comprising:
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establishing expressions quantifying the unwanted dynamics; establishing first constraints bounding an available input to the dynamic system; establishing desired values for the unwanted dynamics at pre-selected values of a physical system characteristic; establishing an error measure based on the differences between the desired values for the unwanted dynamics and the expressions quantifying the unwanted dynamics when evaluated at the pre-selected values of the physical system characteristic; finding a solution which minimizes the error measure and is used to generate the input while still satisfying the first constraints; and controlling the physical system based on the input to the physical system whereby unwanted dynamics are minimized. - View Dependent Claims (127, 128, 129, 130)
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Specification